# Every group admits an initial homomorphism to a pi-powered group

## Statement

Suppose is a group and is a set of primes. There exists a -powered group and a homomorphism of groups such that for any -powered group and any homomorphism , there is a unique homomorphism such that .

We can think of the functor that sends to the group as the **free -powering functor**. It is the left-adjoint functor to the forgetful functor from -powered groups to groups.

### In the localization terminology

The term -*local* is sometimes used to refer to a group that is powered over all the primes **not** in . The functor described above is, in that context, termed the -*localization* functor. By we mean the complement of in the set of all primes.

## Related facts

- The free powered group for a set of primes can be thought of as being obtained from the abstract free group by applying the free -powering functor.

### Embeddability results

We are interested in cases where the canonical homomorphism from the group to its -powering is injective, and in related questions, some of which are explored below.

Group assumption on both the starting group and the big group | Divisibility/powering/torsion assumption on the starting group | Divisibility/powering/torsion assumption on the big group | Is this always possible? | Proof |
---|---|---|---|---|

nilpotent group | none | divisible group | No | nilpotent group need not be embeddable in a divisible nilpotent group |

nilpotent group | -torsion-free group (equivalently, -powering-injective group) | -powered group | Yes | every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group |

arbitrary group | none | divisible group | Yes | every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group |

arbitrary group | -torsion-free group (equivalently, -powering-injective group) | -powered group | No | Powering-injective group need not be embeddable in a rationally powered group |